Download pdf - End to End Learning and Optimization on Graphs - Bryan Wilder · 2021. 1. 2. · 10/27/2019 Bryan Wilder (Harvard) 4. Data. argmax. Decisions. 𝑥𝑥∈𝑋𝑋. 𝑓𝑓𝑥𝑥,𝜃𝜃

End to End Learning and Optimization on Graphs

Bryan Wilder1, Eric Ewing2, Bistra Dilkina2, Milind Tambe11Harvard University

2University of Southern California

To appear at NeurIPS 2019

110/27/2019 Bryan Wilder (Harvard)

10/27/2019 Bryan Wilder (Harvard) 2

Data Decisions???


Data Decisionsarg max𝑥𝑥∈𝑋𝑋 𝑓𝑓 𝑥𝑥,𝜃𝜃

Standard two stage: predict then optimize

Training: maximize accuracy


Data Decisionsarg max𝑥𝑥∈𝑋𝑋 𝑓𝑓 𝑥𝑥,𝜃𝜃

Standard two stage: predict then optimize

Challenge: misalignment between “accuracy” and decision quality

Training: maximize accuracy


Data Decisions

Pure end to end

Training: maximize decision quality


Data Decisions

Pure end to end

Challenge: optimization is hard



Data Decisions

Decision-focused learning: differential optimization during training

arg max𝑥𝑥∈𝑋𝑋

𝑓𝑓 𝑥𝑥,𝜃𝜃



Data Decisions

Challenge: how to make optimization differentiable?

arg max𝑥𝑥∈𝑋𝑋

𝑓𝑓 𝑥𝑥,𝜃𝜃


Decision-focused learning: differential optimization during training

Relax + differentiate


Forward pass: run a solver

Backward pass: sensitivity analysis via KKT conditions

Relax + differentiate• Convex QPs [Amos and Kolter 2018, Donti et al 2018]• Linear and submodular programs [Wilder, Dilkina, Tambe 2019]• MAXSAT (via SDP relaxation) [Wang, Donti, Wilder, Kolter 2019]• MIPs [Ferber, Wilder, Dilkina, Tambe 2019]

• Monday @ 11am, Room 612


What’s wrong with relaxations?• Some problems don’t have good ones• Slow to solve continuous optimization problem• Slower to backprop through – 𝑂𝑂(𝑛𝑛3)


This work• Alternative: include solver for a simpler proxy problem• Learn a representation that maps hard problem to simple one• Instantiate this idea for a class of graph optimization problems


Graph learning + optimization


Problem classes• Partition the nodes into K disjoint groups

• Community detection, maxcut, …• Select a subset of K nodes

• Facility location, influence maximization, vaccination, …• Methods of choice are often combinatorial/discrete


Approach• Observation: grouping nodes into communities is a good heuristic

• Partitioning: correspond to well-connected subgroups• Facility location: put one facility in each community

• Observation: graph learning approaches already embed into 𝑅𝑅𝑛𝑛


Approach1. Start with clustering algorithm (in 𝑅𝑅𝑛𝑛)

• Can (approximately) differentiate very quickly 2. Train embeddings (representation) to solve the particular problem

• Automatically learning a good continuous relaxation!


ClusterNet Approach


Node embedding (GCN)

K-means clustering Locate 1 facility in

each community

Differentiable K-means


Update cluster centers

Softmax update to node assignments

Forward pass



Backward pass

• Option 1: differentiate through the fixed-point condition 𝜇𝜇𝑡𝑡 = 𝜇𝜇𝑡𝑡+1

• Prohibitively slow, memory-intensive



Backward pass


• Prohibitively slow, memory-intensive • Option 2: unroll the entire series of updates

• Cost scales with # iterations• Have to stick to differentiable operations



Backward pass


• Prohibitively slow, memory-intensive • Option 2: unroll the entire series of updates

• Cost scales with # iterations• Have to stick to differentiable operations

• Option 3: get the solution, then unroll one update• Do anything to solve the forward pass• Linear time/memory, implemented in vanilla pytorch

Differentiable K-meansTheorem [informal]: provided the clusters are sufficiently balanced and well-separated, the Option 3 approximate gradients converge exponentially quickly to the true ones.

Idea: show that this corresponds to approximating a particular term in the analytical fixed-point gradients.


ClusterNet Approach


GCN node embeddings


each community

ClusterNet Approach


GCN node embeddings


each community

Loss: quality of facility assignment

ClusterNet Approach


GCN node embeddings


each community


Differentiate through K-means

ClusterNet Approach


GCN node embeddings


each community


Differentiate through K-means

Update GCN params

Experiments• Learning problem: link prediction• Optimization: community detection and facility location problems• Train GCNs as predictive component


Example: community detection


Observe partial graph Predict unseen edges Find communities

max𝑟𝑟

12𝑚𝑚

�𝑢𝑢,𝑣𝑣∈𝑉𝑉

�𝑘𝑘=1

𝐾𝐾

𝐴𝐴𝑢𝑢,𝑣𝑣 −𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣2𝑚𝑚

𝑟𝑟𝑢𝑢𝑘𝑘𝑟𝑟𝑣𝑣𝑘𝑘

𝑟𝑟𝑢𝑢𝑘𝑘 ∈ 0,1 ∀𝑢𝑢 ∈ 𝑉𝑉, 𝑘𝑘 = 1 …𝐾𝐾

�𝑘𝑘=1

𝐾𝐾

𝑟𝑟𝑢𝑢𝑘𝑘 = 1 ∀𝑢𝑢 ∈ 𝑉𝑉

Example: community detection


max𝑟𝑟

12𝑚𝑚

�𝑢𝑢,𝑣𝑣∈𝑉𝑉

�𝑘𝑘=1

𝐾𝐾

𝐴𝐴𝑢𝑢,𝑣𝑣 −𝑑𝑑𝑢𝑢𝑑𝑑𝑣𝑣2𝑚𝑚

𝑟𝑟𝑢𝑢𝑘𝑘𝑟𝑟𝑣𝑣𝑘𝑘

𝑟𝑟𝑢𝑢𝑘𝑘 ∈ 0,1 ∀𝑢𝑢 ∈ 𝑉𝑉, 𝑘𝑘 = 1 …𝐾𝐾

�𝑘𝑘=1

𝐾𝐾

𝑟𝑟𝑢𝑢𝑘𝑘 = 1 ∀𝑢𝑢 ∈ 𝑉𝑉

• Useful in scientific discovery (social groups, functional modules in biological networks)• In applications, two-stage approach is common

[Yan & Gegory ’12, Burgess et al ‘16, Berlusconi et al ‘16, Tan et al ‘16, Bahulker et al ’18…]

Observe partial graph Predict unseen edges Find communities

Experiments• Learning problem: link prediction• Optimization: community detection and facility location problems• Train GCNs as predictive component• Comparison

• Two stage: GCN + expert-designed algorithm (2Stage)• Pure end to end: Deep GCN to predict optimal solution (e2e)


Results: single-graph link prediction


0

0.2

0.4

0.6

Mod

ular

ityCommunity detection

(higher is better)

ClusterNet 2stage e2e5

7

9

11

Max

dist

ance

Facility location (lower is better)

ClusterNet 2stage e2e

Representative example from cora, citeseer, protein interaction, facebook, adolescent health networks

Results: generalization across graphs


0

0.2

0.4

0.6

Mod

ular

ityCommunity detection

(higher is better)

ClusterNet 2stage e2e5

7

9

Max

dist

ance

Facility location (lower is better)

ClusterNet 2stage e2e

ClusterNet learns generalizable strategies for optimization!

Takeaways• Good decisions require integrating learning and optimization• Pure end-to-end methods miss out on useful structure• Even simple optimization primitives provide good inductive bias

NeurIPS’19 paper, see bryanwilder.github.io Code available at https://github.com/bwilder0/clusternet


https://github.com/bwilder0/clusternet

End to End Learning and Optimization on GraphsSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Relax + differentiateRelax + differentiateWhat’s wrong with relaxations?This workGraph learning + optimizationProblem classesApproachApproachClusterNet ApproachDifferentiable K-meansDifferentiable K-meansDifferentiable K-meansDifferentiable K-meansDifferentiable K-meansClusterNet ApproachClusterNet ApproachClusterNet ApproachClusterNet ApproachExperimentsExample: community detectionExample: community detectionExperimentsResults: single-graph link predictionResults: generalization across graphsTakeaways